Motivated by evident lack of studies on the nonlinear stability of porous functionally graded cylindrical shells under combined mechanical loads, this paper presents an effective analytical approach to investigate the postbuckling behavior of moderately thick functionally graded circular cylindrical shells with porosities subjected to combined action of axial compression and external pressure in thermal environments. Porosities are evenly or unevenly distributed within the functionally graded material (FGM). Due to presence of porosities, effective properties of the FGM are determined using a modified version of linear rule of mixture. Formulations are based on first order shear deformation theory taking into account von Kármán-Donnell non-linearity. Analytical solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is adopted to derive closed-form results of buckling loads and nonlinear load-deflection relations. Parametric studies are carried out to evaluate the effects of material and geometry properties, preexisting mechanical loads and temperature, distribution type and volume fraction of porosity on the postbuckling load carrying capacity of porous FGM cylindrical shells.

In this paper, a new improved upwind technique for the moving least squares (MLS) method is proposed for solving convection dominated diffusion problems. The MLS approximation reconstructs a given function from randomly distributed data, which has been successfully applied to many meshless methods. To reduce spurious oscillations and improve accuracy, an upwind strategy with a limiter is designed for the MLS method. The method allows the reconstruction process to switch between different neighborhoods, several numerical examples are tested to confirm the feasibility of the proposed method. For the method, the convection term is approximated on the upwind recomstruction template and a limiter is used while the diffusion term is still approximated by a central reconstruction template. Numerical experiments are conducted to show the superiority and effectiveness of the high-resolution upwind MLS with limiter under a variety of convection dominated diffusion problems, and comparisons are made to see the advantages over some other methods.

In this paper, the stability of the regularized lattice Boltzmann method (RLBM) for convection-diffusion equations is studied. First, it is shown that the evolution equation of RLBM can be transformed into two macroscopic difference schemes, one is explicit and the other is implicit. Compared with the traditional mesoscopic evolution equation, the macro evolution equation has simpler form, lower order of growth matrix, higher computational efficiency and less memory. Then, for the linear convection-diffusion equations without the source term, we use Fourier analysis to prove that the three schemes have exactly the same stability region. For the one-dimensional diffusion equation, we have proved the theoretical stability. For the D1Q3 model, D2Q5 model and D2Q9 model, we have studied the numerical stability. Our results have also been compared with related work by others, which shows they match well. Finally, we conducted some numerical tests to verify that our stability results are credible.

Nonlinear problems widely exist in many aspects of the natural field. The nonlinear situation makes it difficult for most existing solvers to deal with. Therefore, constructing an efficient and accurate solver is a challenge. In this paper, a Legendre spectral method is developed for the nonlinear Volterra integro-differential equation. The error analysis is also provided to justify the spectral rate of convergence for the errors of approximate solution and approximate derivative decay exponentially in both the $L^2$ norm and the infinity norm. In the end, numerical results are displayed to confirm the effectiveness of the Legendre spectral analysis.

In this paper, we develop and analyze a weak Galerkin (WG) finite element method for solving a $H({\rm curl})$-elliptic problem. With the aid of the weak curl operator and a stabilizer term, we first design a WG discretization. Then, by using an auxiliary problem and establishing an error equation, we achieve the optimal order error estimates in both the energy norm and $L^2$ norm for the WG method. At last, we report some numerical experiments to confirm the theoretical results.

In the paper, we consider the coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the newly proposed quadratic auxiliary variable method. Therefore, we develop the multiple quadratic auxiliary variable approach to deal with coupled systems and construct high-accuracy structure-preserving schemes for the equation. To fix the idea, we first apply the multiple quadratic auxiliary variable approach to the equation and obtain an equivalent system that possesses the original energy and mass. Then, a family of high-accuracy structure-preserving schemes that can conserve the mass and energy is derived by applying the Gauss collocation method and sine pseudo-spectral method to approximate the system in time and space. The given schemes have high-accuracy in time and can both inherit the mass and Hamiltonian energy of the system. Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.

In this paper, a hybrid Immersed-Boundary/Body-Fitted-Grid method is extended to simulate the complex fluid-structure-interaction (FSI) problems involved in the flows through a bileaflet mechanical heart valve (BMHV). The background grid is discretized with a body-fitted gird while the freely-rotating leaflets are treated by an immersed boundary method named the local domain-free-discretization (DFD) method. For simulation of turbulence involved in BMHV flows, the wall-modelled large eddy simulation is implemented in the DFD framework. In order to address the instability issue associated with low-mass structures, the strong coupling (SC) strategy is employed. To accelerate the convergence of SC-FSI sub-iteration, we adopt the under-relaxion scheme with an optimal relaxion coefficient and Hamming’s 4th-order predictor-corrector scheme. To validate the SC schemes for low-mass structures, numerical experiments on the vortex induced vibration of an elastically mounted cylinder are carried out. Finally, the present method is applied to simulate and investigate the pulsatile flows through a BMHV at physiologic conditions, in which complex fluid-structure interactions are involved. The computed results agree well with the published numerical or experimental data.

This paper investigates a novel nonlinear singular fractional SI model with the $Φ_p$ operator and the Mittag-Leffler kernel. The initial investigation includes the existence, uniqueness, boundedness, and non-negativity of the solution. We then establish Hyers-Ulam stability for the proposed model in Banach space. Optimal control analysis is performed to minimize the spread of infection and maximize the population of susceptible individuals. Finally, the theoretical results are supported by numerical simulations.

In this paper, a first-order penalty finite element method for the 2D/3D unsteady incompressible thermomicropolar fluid (UITF) equations is considered, which combines the advantage of the penalty method, first-order backward Euler scheme, and implicit or explicit iteration for the nonlinear terms to get a decoupled temporal evolution procedure, which only needs to solve a series of elliptic subproblems. Theoretically, the stability analysis and optimal error estimates of the temporal semi-discrete method are deduced. Furthermore, the classical MINI element pairs are adopted in concrete spatial discrete, and the feasibility of the method is verified by numerical experiments.

In this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of plates made of functionally graded material is presented. It is assumed that the properties of the functionally graded plates are in the form of the simple power law scheme and continuous along the thickness, under thermal load and incident oblique plane sound wave as well as the first-order shear deformation theory. For this purpose, first, using Hamilton’s principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the nonlinear von Kármán strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to Duffing equation. Then, using the homotopy analysis method, the equation of the transverse movement of the plate is solved semi-analytically to obtain the nonlinear frequencies. Finally, the nonlinear vibration and acoustic response of functionally graded plates are studied by considering the variation of the important parameters such as aspect ratio, dimensionless amplitude, volume fraction power of functionally graded material, external acoustic pressure, incidence and azimuthal angles, temperature changes, phase portrait, sound transmission loss, velocity and average mean square velocity of drive point and sound power level of the functionally graded plate. Results show increasing the incidence angle leads increase in hardening effects and sound transmission loss, but growing the azimuthal angle does not have much effect on the frequency-response and sound transmission loss in the absence of the external mean flow. Also, increasing temperature changes lead to decrease in hardening effects and sound transmission loss.